Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

===Setup===

Let's follow the guidelines of the [[SSC/SynopsisStyleSheet|variational principle]].  Instead of starting with the form of the LAWE given above, namely,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

we will start with a form that is more amenable to the variational principle, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{dr_0}\biggl[ r_0^4 \gamma P_0 ~\frac{dx}{dr_0} \biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 + (3\gamma - 4) r_0^3 \frac{dP_0}{dr_0} \biggr] x \, .
</math>
  </td>
</tr>
</table>
</div>

<table border="1" cellpadding="8" align="center" width="85%"><tr><td align="left">
<font color="red">'''ASIDE:'''</font>  Let's show that these two expressions are equivalent.  Remembering that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP_0}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- g_0 \rho_0 = - \biggl(\frac{GM_r}{r_0^2}\biggr)\rho_0 \, ,</math>
  </td>
</tr>
</table>
the second expression becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_0^4 \gamma P_0 ~\frac{d^2x}{dr_0^2} 
+ ~\gamma \frac{dx}{dr_0}\biggl[ 4r_0^3 P_0 - r_0^4 g_0 \rho_0\biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 - (3\gamma - 4) r_0^3 g_0 \rho_0\biggr] x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_0^4 \gamma P_0 \biggl\{\frac{d^2x}{dr_0^2} 
+ \biggl[ \frac{4}{r_0} - \frac{g_0 \rho_0}{P_0} \biggr] \frac{dx}{dr_0}
+\frac{\rho_0}{\gamma P_0 }\biggl[ \omega^2 - (3\gamma - 4) \frac{g_0}{r_0}  \biggr] x 
\biggr\}\, .
</math>
  </td>
</tr>
</table>
Hence, we must multiply the first expression through by <math>~r_0^4 \gamma P_0</math> in order to obtain the second expression.
</td></tr></table>


From [[#Foundation|above]], we realize that multiplying the second expression through by <math>~(K_c/G)\rho_c^{-4 / 5}</math> gives,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_0^4\gamma P_0 \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(r^*)^4 [K_c^{1 / 2}/(G^{1 / 2} \rho_c^{2/5}) ]^4 \gamma P^* [K_c\rho_c^{6/5}] \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K_c^{3}}{ G^{2} \rho_c^{2/5}} \biggr]  (r^*)^4 \gamma P^* \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

That is, multiplying the second expression through by,  <math>~(K_c/G)\rho_c^{-4 / 5} \cdot G^2 \rho_c^{2 / 5}/K_c^3 = G/(K_c^2 \rho_c^{2 / 5})</math> , should give a desirable, totally dimensionless version of the LAWE. Remembering that,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0}{\rho_c}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{[K_c^{1/2}/(G^{1/2}\rho_c^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{K_c\rho_c^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_c^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dP^*}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{M^*_r \rho^*}{(r^*)^2} </math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K_c^5}{G^3}\biggr]^{1 / 2}</math>
  </td>
</tr>

</table>
</div>
let's try it.

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G K_c^{-2} \rho_c^{-2 / 5} \biggl\{
\frac{d}{dr_0}\biggl[ r_0^4 \gamma P_0 ~\frac{dx}{dr_0} \biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 + (3\gamma - 4) r_0^3 \frac{dP_0}{dr_0} \biggr] x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G K_c^{-2} \rho_c^{-2 / 5} \biggl\{ \biggl( \frac{K_c^2 \rho_c^{2/5}}{G} \biggr)
\frac{d}{dr^*}\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+\biggl[ \omega^2 \biggl(\frac{K_c^2}{G^2 \rho_c^{8/5}}\biggr) \rho_c \rho^* (r^*)^4 + (3\gamma - 4)\biggl(\frac{K_c }{G \rho_c^{4/5}}\biggr)K_c\rho_c^{6/5} (r^*)^3 \frac{dP^*}{dr^*} \biggr] x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{dr^*}\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] x \, .
</math>
  </td>
</tr>
</table>

Now, guided by the [[SSC/SynopsisStyleSheet#Stability|accompanying summary]], if we multiply through by <math>~4\pi x dr^*</math> and integrate over the entire volume, we obtain the ''governing variational relation'', namely,
<!-- OLD VERSION; IGNORE

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_0^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 4\pi  x(r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr]_0^{R^*} 
- \int_0^{R^*}4\pi  \biggl[ (r^*)^4 \gamma P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
+ \int_0^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4  \biggr] 4\pi x^2 dr^*
- \int_0^{R^*}\biggl[ (3\gamma - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \int_0^{R^*}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma 4\pi (r^*)^2 P^*  dr^*
+ \int_0^{R^*} (3\gamma - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{R^*} 
+ \int_0^{R^*} 4\pi \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4   x^2 dr^* \, .
</math>
  </td>
</tr>
</table>
OLD VERSION; IGNORE-->

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma_c P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_0^{r^*_\mathrm{core}}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma_c - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\int_{r^*_\mathrm{core}}^{R^*}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma_e P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_{r^*_\mathrm{core}}^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma_e - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 4\pi  x(r^*)^4 \gamma_c P^* ~\frac{dx}{dr^*} \biggr]_0^{r^*_\mathrm{core}} 
- \int_0^{r^*_\mathrm{core}}4\pi  \biggl[ (r^*)^4 \gamma_c P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
- \int_0^{r^*_\mathrm{core}}\biggl[ (3\gamma_c - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ 4\pi  x(r^*)^4 \gamma_e P^* ~\frac{dx}{dr^*} \biggr]_{r^*_\mathrm{core}}^{R^*} 
- \int_{r^*_\mathrm{core}}^{R^*}4\pi  \biggl[ (r^*)^4 \gamma_e P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
- \int_{r^*_\mathrm{core}}^{R^*}\biggl[ (3\gamma_e - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
+ \int_{0}^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4  \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma_c 4\pi (r^*)^2 P^*  dr^*
+ \int_0^{r^*_\mathrm{core}} (3\gamma_c - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma_c P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{r^*_\mathrm{core}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \int_{r^*_\mathrm{core}}^{R^*}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma_e 4\pi (r^*)^2 P^*  dr^*
+ \int_{r^*_\mathrm{core}}^{R^*} (3\gamma_e - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma_e P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_{r^*_\mathrm{core}}^{R^*} 
+ \int_{0}^{R^*} 4\pi \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4   x^2 dr^* \, .
</math>
  </td>
</tr>
</table>


Energy Normalization:

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\gamma - 1)dU_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi r^2 P dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl[ \frac{K_c^{1 / 2}}{G^{1 / 2} \rho_c^{2 / 5}} \biggr]^3 K_c \rho_c^{6 / 5} (r^*)^2 P^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl[ \frac{K_c^{5}}{G^{3}} \biggr]^{1 / 2} (r^*)^2 P^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K_c^{5}}{G^{3}} \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dW_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{GM^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^* \biggl[ \frac{K_c}{G\rho_c^{4 / 5}} \biggr]\rho_c \biggl[ \frac{ K_c^{3 / 2} }{G^{3 / 2}\rho_c^{1 / 5}  } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{M^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^*  E_\mathrm{norm}
</math>
  </td>
</tr>
</table>

<span id="VariationPrincipleRelation">Hence, the ''dimensionless'' governing variational relation becomes,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_0^{R^*} \biggl( \frac{2\pi}{3}\biggr)\sigma_c^2(r^*)^2   x^2 dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \int_0^{r^*_\mathrm{core}}  \gamma_c(\gamma_c - 1) x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- \int_0^{r^*_\mathrm{core}} (3\gamma_c - 4)  x^2 dW^*_\mathrm{grav}
+ \biggl[ 4\pi  x^2 (r^*)^3 \gamma_c P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{r^*_\mathrm{core}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \int_{r^*_\mathrm{core}}^{R^*}  \gamma_e(\gamma_e - 1) x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- \int_{r^*_\mathrm{core}}^{R^*} (3\gamma_e - 4)  x^2 dW^*_\mathrm{grav}
+ \biggl[ 4\pi  x^2 (r^*)^3 \gamma_e P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_{r^*_\mathrm{core}}^{R^*} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
+ 4\pi  x_i^2 (r_\mathrm{core}^*)^3 \gamma_c P_i^* \biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 \gamma_e P_i^* \biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^*\biggl[ \gamma_c \biggl\{ \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}  - \gamma_e \biggl\{ \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^*\biggl[ 3(\gamma_e - \gamma_c) \biggr] \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dM^*_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi (r^*)^2 \rho^* dr^*  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dU^*_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\gamma-1)} \biggl[4\pi (r^*)^2 P^*  dr^* \biggr] = \biggl[ \frac{2}{3(\gamma - 1)} \biggr]dS^*_\mathrm{therm}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{M^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^*  \, .</math>
  </td>
</tr>
</table>

Or, for inclusion in our accompanying ''[[SSC/SynopsisStyleSheet#Bipolytropes|Tabular Overview]]'',
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2  dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^*  \, .
</math>
  </td>
</tr>
</table>